First of all, for the equation on the third line, a minus sign is missing in front of the 'iw'.
Secondly, it seems to me that the direction of the induced EMF, as shown by the red arrows on the dashed red ellipse, should be the way around (i.e., should be clockwise). The reason is as follows. If the time-varying magnetic flux is increasing in the direction of the black arrow, then the induced EMF should be in such a direction that generates a secondary magnetic field that is in the opposite direction of the increasing primary magnetic field. Therefore, the induced EMF should be clockwise.
Could you please check the direction of the induced EMF?
The following screenshot was taken from this webpage (https://em.geosci.xyz/content/maxwell3_fdem/circuitmodel_for_eminduction/index.html).
First of all, for the equation on the third line, a minus sign is missing in front of the 'iw'.
Secondly, it seems to me that the direction of the induced EMF, as shown by the red arrows on the dashed red ellipse, should be the way around (i.e., should be clockwise). The reason is as follows. If the time-varying magnetic flux is increasing in the direction of the black arrow, then the induced EMF should be in such a direction that generates a secondary magnetic field that is in the opposite direction of the increasing primary magnetic field. Therefore, the induced EMF should be clockwise.
Could you please check the direction of the induced EMF?